Problem 22 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$5 x^{2}-15 x-50$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(5x^2-15x-50\) is \(5(x-5)(x+2)\).

Step 1: Identify the Greatest Common Factor

Look for a common factor in each term of the polynomial. Here, each coefficient is divisible by 5. So, 5 is the greatest common factor (GCF).

Step 2: Factor the GCF Out

Factor out the GCF from each term of the polynomial. Applying it here, \(5(x^2 - 3x - 10)\) would be the polynomial after factoring out 5.

Step 3: Factor the Quadratic Expression

Now, need to factor the quadratic expression, \(x^2 - 3x - 10\). This can be done by finding two numbers that multiply to -10 and add up to -3. The numbers that fit this criterion are -5 and 2. So, the factored form of the quadratic expression would be \((x - 5)(x + 2)\).

Step 4: Write the Completely Factored Form

Finally, include the GCF with the factored form of the quadratic expression in the final answer. Therefore, the completely factored form of the polynomial is \(5(x - 5)(x + 2)\).

Step 5: Check the Solution

This involves either multiplying the factors to ensure it equals the original polynomial or plot the original polynomial and the factored one on a graph to see if they intersect at the same points. In this case, it's observed that by multiplying \(5(x - 5)(x + 2)\), the original polynomial \(5x^2-15x-50\) is obtained. Thus, the solution has been checked and confirmed.

Key Concepts

Greatest Common Factor

When beginning the process of factoring polynomials, the first step often involves identifying the Greatest Common Factor (GCF). This is the largest number that divides all the coefficients of a polynomial without leaving a remainder. Finding the GCF simplifies the problem and makes further factoring easier.

In the example of the polynomial

• \(5x^2 - 15x - 50\), notice how each term shares something in common, as 5 divides 5, 15, and 50.
• Thus, 5 is the GCF, which is factored out from every term, reducing the expression to \(5(x^2 - 3x - 10)\).

This process streamlines the polynomial, setting the stage for simpler factoring of more complex parts of the expression. Keeping an eye out for the GCF right from the start is crucial because it is always the first step towards complete factorization.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, generally written in the form \(ax^2 + bx + c\). They are called 'quadratic' because 'quad' indicates square, the highest power of the variable.

Factoring quadratic expressions involves rewriting them as the product of two binomials. For the expression \(x^2 - 3x - 10\), the task is to find two numbers that both multiply to give -10 (the constant term) and add to give -3 (the coefficient of the linear term).

The trick is to think about pairs of factors of the constant term -10, such as:

• (1 and -10)
• (-1 and 10)
• (2 and -5).

Among these, -5 and 2 meet the required conditions, since -5 + 2 = -3 and -5 * 2 = -10. Thus, the quadratic expression simplifies into the factored form \((x - 5)(x + 2)\). Recognizing these pairs and manipulating numbers is key to mastering the factoring of quadratics.

Factored Form

Once you've successfully factored an expression by identifying the GCF and breaking down any quadratic expressions, the next step is putting it all together in the factored form. This comprehensive form shows the polynomial as a product of its factors and is crucial for solving polynomial equations by setting each factor to zero.

For our polynomial \(5x^2 - 15x - 50\), the factorization process yields the final expression: \(5(x - 5)(x + 2)\). The GCF (5) is combined with the factored quadratic expression to represent the complete factorization.

This form is not only useful for computations but also provides insights into the roots of the polynomial. In this case, setting each part equal to zero gives solutions (roots) at \(x = 5\) and \(x = -2\). Each step in the transition to the factored form offers insights beyond mere arithmetic; they reveal deeper properties of the polynomial structure.